Determine whether the following sets are subspaces of r 3 under the operations of addition and scalar multiplication defined on r 3 . justify your answers. (a) w 1 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : a 1 = 3 a 2 and a 3 = − a 2 } (b) w 2 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : a 1 = a 3 + 2 } (c) w 3 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : 2 a 1 − 7 a 2 + a 3 = 0 } (d) w 4 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : a 1 − 4 a 2 − a 3 = 0 } (e) w 5 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : a 1 + 2 a 2 − 3 a 3 = 1 } (f) w 6 = { ( a 1 , a 2 , a 3 ) ∈ r 3 : 5 a 2 1 − 3 a 2 2 + 6 a 2 3 = 0 }

i would be happy to . most people on would also love to i'm sure. if you just want someone who has done algebra before and want a little more personal relationship than just begging strangers for , i can certainly be of service. i do have college myself so i might not be 24/7.

the image is given below

since the options are not given so we draw it from the equation

the given equation is

y = (x-1) (x+4)

y = y² + 3x - 4

the above equation is a quadratic equation which will have two roots. if we draw the graph for it, it will look like this

  b:   ∠x = 28.6°, ∠y = 31.2°, ∠z = 120.2°

step-by-step explanation:

this is one of those questions where "test taking skill" is all you need. the only answer that has angles that sum to 180° is the one shown above. the remaining choices do not describe the angles of a triangle.

if you want to do more work than simply adding the offered numbers, you can also check them against the law of sines. it should be true that

  x/sin(x) = y/sin(y) = z/sin(z)

and all of these ratios must be greater than 15, the longest side length.

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if you want to solve the triangle from scratch, i'd suggest solving for the largest angle using the law of cosines.

  z² = x² +y² -2xy·cos(z)

  z = arccos((x² +y² -z²)/(2xy)) = arccos(-75.11/149.4) = 120.182°

now you can find y from the law of sines

  y = arcsin(y/z·sin(z)) = 31.242°